Problem 1: Vacancies and free energy [18 points} In a crystalline solid. thermal uctuations can cause an atom to spontaneously jump out of its lattice site and cram itself into an interstitial (I) site, leaving a vacancy {V} behind. The number of possible sites for an interstitial [or vacancy} is about equal to the number N of atoms in the solid. An activation energy E is required to form a single IV pair, regardless of the nal I and V positions. At a given temperature. a vacancyinterstitial {VI) pairs are created among the N lattice sites of a crystalline solid. Assume N 23:? n '3} l. i . . i . i I C I- . i . .I . .1. . - I' i . I . I I I :I I I I. V {a} Derive an expression for the entropy of n interstitialvacancy pairs among N lattice sites. Be aware that once created1 the members of the pair can wander anywhere [the LEFT pair does not stay together and so their locations are random]. Use the Stirling Approx. 1with N 33> 70. 23> l to simplify your answer. [Hint The multiplicity of accessible microstates is 2 = (2:91;, where if]; and f 1;: are the mul tiplicities of interstitials and vacancies respectively. As an intermediate step. you should get lnll; = n + nlnNnlnn.] {b} 1Write an expression for the free energg,r of the system at a temperature T as a function of the number n of vacancies. (c) Minimize the free energy with respect to 11 to derive an expression for the equilibrium number of 117 pairs 71 = nfT}. Simplify your expression as much as possible